晶格杨-米尔斯-希格斯动力学及其应用
Langevin dynamics of lattice Yang-Mills-Higgs and applications
AI总结：
本文研究了晶格杨-米尔斯-希格斯模型的兰格-恩斯汀动力学，证明了其指数 ergodicity 和相关衰减，扩展了纯晶格杨-米尔斯结果到包含耦合希格斯场的情况。
摘要：
本文研究了各种晶格形式的杨-米尔斯-希格斯模型的兰格-恩斯汀动力学，其中包含反杨-米尔斯耦合参数β和希格斯参数κ。希格斯分量要么是取值于紧致目标空间的有界场，要么是取值于向量空间的无界场，后一种情况还包含一个希格斯质量参数m。我们研究了当(β,κ)在第一种情况下较小或(β,κ/m)在第二种情况下较小的 regime。我们通过函数不等式证明了动力学在整个晶格上的指数 ergodicity。我们建立了对广泛可观测值的指数相关衰减，即无限体积测度表现出严格正的质量间隙。此外，当希格斯场的目标空间为紧致时，适当缩放的可观测值在大N极限下表现出因子化的相关性。这些结果扩展了之前[1]关于纯晶格杨-米尔斯的结果到包含耦合希格斯场的情况。不同于纯晶格杨-米尔斯中场总是有界的情况，在耦合希格斯分量无界的情况下，对其行为的控制更加困难，需要新的技术。我们的方法涉及分解论证和对相关性的精细分析以有效控制无界希格斯分量。
英文摘要：
In this paper, we investigate the Langevin dynamics of various lattice formulations of the Yang--Mills--Higgs model, with an inverse Yang--Mills coupling $β$ and a Higgs parameter $κ$. The Higgs component is either a bounded field taking values in a compact target space, or an unbounded field taking values in a vector space in which case the model also has a Higgs mass parameter $m$. We study the regime where $(β,κ)$ are small in the first case or $(β,κ/m)$ are small in the second case. We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. We establish exponential decay of correlations for a broad class of observables, namely, the infinite volume measure exhibits a strictly positive mass gap. Moreover, when the target space of the Higgs field is compact, appropriately rescaled observables exhibit factorized correlations in the large $N$ limit. These extend the earlier results \cite{SZZ22} on pure lattice Yang--Mills to the case with a coupled Higgs field. Unlike pure lattice Yang--Mills where the field is always bounded, in the case where the coupled Higgs component is unbounded, the control of its behavior is much harder and requires new techniques. Our approach involves a disintegration argument and a delicate analysis of correlations to effectively control the unbounded Higgs component.
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